Recap Lecture 3
RE, Recursive definition of RE, defining
languages by RE, { x}*, { x}+, {a+b}*,
Language of strings having exactly one aa,
Language of strings of even length, Language
of strings of odd length, RE defines unique
language (as Remark), Language of strings
having at least one a, Language of strings
havgin at least one a and one b, Language of
strings starting with aa and ending in bb,
Language of strings starting with and ending
in different letters.
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Task
Determine the RE of the language, defined over
Σ={a, b} of words beginning with a.
Solution:
The required RE may be a(a+b)*
Determine the RE of the language, defined over
Σ={a, b} of words beginning with and ending in
same letter.
Solution:
The required RE may be (a+b)+a(a+b)*a+b(a+b)*b
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Task Continued …
Determine the RE of the language, defined over
Σ={a, b} of words ending in b.
Solution:
The required RE may be
(a+b)*b.
Determine the RE of the language, defined over
Σ={a, b} of words not ending in a.
Solution: The required RE may be
(a+b)*b + Λ Or ((a+b)*b)*
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An important example
The Language EVEN-EVEN :
Language of strings, defined over Σ={a, b}
having even number of a’s and even
number of b’s. i.e.
EVEN-EVEN = {Λ, aa, bb, aaaa,aabb,abab,
abba, baab, baba, bbaa, bbbb,…} ,
its regular expression can be written as
(aa+bb+(ab+ba)(aa+bb)*(ab+ba))*
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Note
 It is important to be clear about the
difference of the following regular
expressions
r1=a*+b*
r2=(a+b)*
Here r1 does not generate any string of
concatenation of a and b, while r2
generates such strings.
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Equivalent Regular Expressions
Definition:
Two regular expressions are said to be
equivalent if they generate the same language.
Example:
Consider the following regular expressions
r1= (a + b)* (aa + bb)
r2= (a + b)*aa + ( a + b)*bb then
both regular expressions define the language of
strings ending in aa or bb.
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Note
 If r1
=(aa + bb) and r2=( a + b) then
1. r1+r2 =(aa + bb) + (a + b)
2. r1r2 =(aa + bb) (a + b)
=(aaa + aab + bba + bbb)
3. (r1)* =(aa + bb)*
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Regular Languages
 Definition:
The language generated by any regular
expression is called a regular language.
It is to be noted that if r1, r2 are regular
expressions, corresponding to the languages L1
and L2 then the languages generated by r1+ r2,
r1r2( or r2r1) and r1*( or r2*) are also regular
languages.
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Note
 It is to be noted that if L1 and L2 are expressed by r1and
r2, respectively then the language expressed by
1) r1+ r2, is the language L1 + L2 or L1 U L2
2) r1r2, , is the language L1L2, of strings obtained by
prefixing every string of L1 with every string of L2
3) r1*, is the language L1*, of strings obtained by
concatenating the strings of L, including the null string.
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Example
 If r1=(aa+bb) and r2=(a+b) then the language of
strings generated by r1+r2, is also a regular language,
expressed by (aa+bb)+(a+b)
 If r1=(aa+bb) and r2=(a+b) then the language of
strings generated by r1r2, is also a regular language,
expressed by (aa+bb)(a+b)
 If r=(aa+bb) then the language of strings generated by
r*, is also a regular language, expressed by (aa+bb)*
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All finite languages are regular.
Example:
Consider the language L, defined over Σ={a,b},
of strings of length 2, starting with a, then
L={aa, ab}, may be expressed by the regular
expression aa+ab. Hence L, by definition, is a
regular language.
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Note
It may be noted that if a language contains
even thousand words, its RE may be expressed,
placing ‘ + ’ between all the words.
Here the special structure of RE is not
important.
Consider the language L={aaa, aab, aba, abb,
baa, bab, bba, bbb}, that may be expressed by
a RE aaa+aab+aba+abb+baa+bab+bba+bbb,
which is equivalent to (a+b)(a+b)(a+b).
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Introduction to Finite
Automaton
 Consider the following game board that contains
64 boxes
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Finite Automaton Continued …
There are some pieces of paper. Some are of
white colour while others are of black color. The
number of pieces of paper are 64 or less. The
possible arrangements under which these pieces
of paper can be placed in the boxes, are finite.
To start the game, one of the arrangements is
supposed to be initial arrangement. There is a
pair of dice that can generate the numbers
2,3,4,…12 . For each number generated, a
unique arrangement is associated among the
possible arrangements.
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Finite Automaton Continued …
It shows that the total number of transition
rules of arrangement are finite. One and more
arrangements can be supposed to be the
winning arrangement. It can be observed that
the winning of the game depends on the
sequence in which the numbers are generated.
This structure of game can be considered to be
a finite automaton.
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Defining Languages (continued)…
 Method 4 (Finite Automaton)
Definition:
A Finite automaton (FA), is a collection of the
followings
1) Finite number of states, having one initial and some
(maybe none) final states.
2) Finite set of input letters (Σ) from which input
strings are formed.
3) Finite set of transitions i.e. for each state and for
each input letter there is a transition showing how
to move from one state to another.
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Example
 Σ = {a,b}
 States: x, y, z where x is an initial state and z is final
state.
 Transitions:
1. At state x reading a go to state z,
2. At state x reading b go to state y,
3. At state y reading a, b go to state y
4. At state z reading a, b go to state z
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Example Continued …
These transitions can be expressed by the
following table called transition table
Old States
xy
z+
New States
Reading a
z
y
z
Reading b
y
y
z
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Note
It may be noted that the information of an FA,
given in the previous table, can also be depicted
by the following diagram, called the transition
diagram, of the given FA
a,b
y
b
x–
a,b
a
Z+
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Remark
The previous transition diagram is an FA
accepting the language of strings, defined over
Σ={a, b}, starting with a. It may be noted
that this language may be expressed by the
regular expression
a (a + b)*
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Summing Up
Regular expression of EVEN-EVEN language,
Difference between a* + b* and (a+b)*,
Equivalent regular expressions; sum, product
and closure of regular expressions; regular
languages, finite languages are regular,
introduction to finite automaton, definition of
FA, transition table, transition diagram
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